Ch 24 pages 625-627 - PowerPoint PPT Presentation

Lecture 5 – Transport pr operties of gases Ch 24 pages 625-627 Summar y of lecture 4 We can use Maxwell-Boltzmann distribution to calculate the average values for any property that depends on speed, e.g. Summar y of lecture 4

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The motion of individual molecules in gases or fluids allows us to explore microscopic aspects of diffusion

Let us for consider the motions a solute molecule (e.g. DNA or protein under electrophoresis or centrifugation) surrounded by solvent molecules with which it collides. If the trajectory (pathway) of an individual solvent molecule is traced out, it appears to be a random walk, much like the zig-zag motion of the molecules in an ideal gas we described in the last lecture

Other molecules will be displaced in other directions. For every solute molecule displaced x as shown, there will be another solute molecule with a displacement in the opposite direction –x. The displacements will all average to zero, that is

However, the mean-squared displacement is non-zero. The random-walk description allows us to calculate this averages

Many processes in nature including the motion of a drunk man on the way home, Brownian particles, motions of gas molecules, and the average shape of a linear polymer, can be treated as “random walk” problems

The question to be addressed is: what is the average distance between the beginning and end of a path of N random steps of length l?

These expressions are stating that, after N steps, one is (on average)

away from where it started. The mean displacement is always 0, because the probability of moving back or forward is the same. For any random walk, the root mean square displacement is the microscopic unit displacement times the square root of the total number of hops. This is a fundamental property of random walk.

Thus, although the average speed of a particle in a gas is very high (>100 m/s) it takes a very long time for molecules to diffuse in a gas because the root mean square displacement is inversely proportional to the rate at which molecules collide

We have expressed the probability in terms of a number displacement, but it is useful to do so in terms of the net distance displacement x=lm where l is the unit displacement (the length of each step taken). Through a simple substitution:

Suppose the number of hops or steps per unit time is N’. Then the number of hops N=N’t. Therefore we can also express this probability in terms of frequency of steps and time: